Let us call a **subgroup** an injective homomorphism between groups.

I warn the reader that asubgroupdesignates here an inclusion $(H \subset G)$, not $H$ alone.

A subgroup $H \subset G$ is **maximal** if for all intermediate subgroups $H \subset K \subset G$, then $K=H$ or $G$.

Let $\sim$ be the **equivalence of subgroups** (defined here).

Note that the maximality is invariant under $\sim$.

Let $n$ be a fixed integer. For each equivalence class of index $n$ maximal subgroups, we choose a representative $(H \subset G)$ with $G$ of minimal order. Let $R_{n}$ be the set of all these representatives.

**Interrelated questions :**

- Does $R_{n}$ is a finite set ?
- $\forall (H \subset G) \in R_{n}$, is $ord(G)$ bounded ?
- Is $G$ always a finite group ? A counter-example ?

**Original motivation**: What's the list of all the maximal subgroups at index $6$ ?